Laguerre geometry and Dupin hypersurfaces with constant Laguerre curvatures in Rn+1

Abstract

In this work, we present the results studied in Caixeta e Rodrigues [4]. Initially, we studied the geometry of the oriented spheres in Rⁿ⁺¹ based on the work of Cecil [1], and the Laguerre geometry in the Euclidean space, according to the article by Li e Wang [6]. Subsequently, considering Mⁿ ⊂ Rⁿ⁺¹ an oriented hypersurface with r (r ≥ 3)  distinct nonvanishing principal curvatures, we present a characterization obtained by Li e Wang [7], in terms of Laguerre invariants, of Dupin hypersurfaces with constant Laguerre curvatures. We also present the classification result of the Dupin hypersurfaces with constant Laguerre curvaturesproposed by Li e Wang [7], which consists in showing that a Dupin hypersurfaces with constant Laguerre curvatures is Laguerre equivalent to a flat Laguerre isoparametric hypersurface. In a slightly different context from the one studied so far, considering a proper Dupin hypersurfaces of the Euclidean space Rⁿ⁺¹, that admit principal coordinate systems and have n distinct nonvanishing principal curvatures Cezana e Tenenblat [2] presented a characterization of Dupin hypersurfaces in Rⁿ⁺¹, n ≥ 3, with all the distinct principal curvatures and constant Laguerre curvatures, that is parametrized by lines of curvature, in terms of the radius of curvature and their first fundamental form. So, using this result Cezana e Tenenblat [2] showed explicitly all such hypersurfaces that have constant Laguerre curvatures.